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{{ printedBook.courseTrack.name }} {{ printedBook.name }} Two circles are congruent circles if and only if they have the same radius.

Consider two congruent circles $⊙A$ and $⊙B,$ and a point on each one.

Because $⊙A≅⊙B,$ the distance from the center of the circle to a point on the circle is the same for each circle. Therefore, $AC≅BD$ which implies that both circles have the same radius.

Conversely, consider two circles with the same radius.

Next, translate $⊙B$ so that point $B$ maps to point $A.$ The image of $⊙B$ is $⊙B_{′}$ which is a circle centered at $A.$ Since the circles have the same radius, this translation maps $⊙B$ onto $⊙A.$Translate $⊙B$

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Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.